SWA has recently started interviewing applicants without B737 type ratings. I am not a SWA employee but I can say with some certainty that about 30 percent of those who interview don't have types. Even so conditions remain very competitive at SWA, on average only about 3 in 12 are hired.
Most have either very high flight time, are Line Check Airman or both. LCAs that have been called have about 6000 TT and 4000 PIC turbine. Once you're offered a position you have 6 months to attain the rating and only then are you placed in the pool. Wait times for a class are running about 4 months.
Hope this helps.